Exercise 13, § 6

Let $X$ be a topological space whose every point admits a countable fundamental system of neighbourhoods, $Y$ a complete metric space, $f : X \to Y$ a continuous open mapping, $B$ a countable closed subset of $Y$. Show that there exists a continuous section of $f$ over $B$, i.e. a continuous mapping $s$ of $B$ into $X$ such that $f(s(y)) = y$ for all $y \in B$. (Let $C$ be the set of the $y \in B$ such that there exists a neighbourhood V of $y$ in $Y$ and a continuous section of $f$ over $V \cap B$. Show that the closed set $B - C$ cannot admit isolated points, and conclude that $B = C$ by using the Baire theorem).